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Cumulative HigherOrder Logic as a Foundation for Set Theory
"... The systems K of transnite cumulative types up to are extended to systems K 1 that include a natural innitary inference rule, the socalled limit rule. For countable a semantic completeness theorem for K 1 is proved by the method of reduction trees, and it is shown that every model of K 1 ..."
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The systems K of transnite cumulative types up to are extended to systems K 1 that include a natural innitary inference rule, the socalled limit rule. For countable a semantic completeness theorem for K 1 is proved by the method of reduction trees, and it is shown that every model of K 1 is equivalent to a cumulative hierarchy of sets. This is used to show that several axiomatic rstorder set theories can be interpreted in K 1 , for suitable . Keywords: cumulative types, innitary inference rule, logical foundations of set theory. MSC: 03B15 03B30 03E30 03F25 1 Introduction The idea of founding mathematics on a theory of types was rst proposed by Russell [20] (foreshadowed already in [19]), and subsequently implemented by Whitehead and Russell [26]. The formal systems presented in these works were later simplied and cast into their modern shape by Ramsey [18]. Godel [9] and Tarski [25] were the rst to restrict the type structure to types of unary predi...